by Robert S. Pindyck and Julio J. Rotemberg September 1982 WP #1351-82
Research leading to this paper was supported by the National Science Foundation under Grant No. SES-8012667 to R. S. Pindyck, and Grant No. to J. J. Rotemberg. That financial support is greatly appreciated. We also want to thank Ernst Berndt and Dale Jorgenson for providing us their data, George Pennacchi for his superb research assistance, and Lawrence Summers for helpful conversations.
"---"---Dynamic Factor Demands Under Rational Expectations by Robert S. Pindyck and Julio J. Rotemberg ABSTRACT
This paper presents a dynamic model of the industrial demands for structures, equipment, and blue- and white-collar labor. Our approach is consistent with producers holding rational expectations and optimizing dynam-ically in the presence of adjustment costs, yet it permits generality of func-tional form regarding the technology. We represent the technology by a
translog input requirement function that specifies the amount of blue-collar labor (a flexible factor) the firm must hire to produce a level of output given its quantities of three quasi-fixed factors that are subject to adjust-ment costs: non-production (white-collar) workers, equipment, and structures. A complete description of the production structure is obtained by simultaneously estimating the input requirement function and three stochastic Euler equations. We apply an instrumental variable technique to estimate these equations using aggregate data for U.S. manufacturing. We find that as a fraction of total expenditures, adjustment costs are small in total but large on the margin, and that they differ considerably across quasi-fixed factors. We also present short- and long-run elasticities of factor demands.
Understanding the way in which tax changes, changes in relative factor prices, and changes in aggregate output affect investment and employment over time requires a model of the production structure that incorporates dynamic adjustment of "quasi-fixed" factors, i.e. a dynamic model of factor demands. This paper develops such a model in a way that is consistent with rational
expectations and dynamic optimization in the presence of adjustment costs, while allowing for generality of functional form.
Dynamic factor demand models are certainly not new to the literature. The "flexible accelerator" and related models of investment demand have been
widely used in empirical applications, although they are generally based on ad hoc descriptions of the dynamic adjustment process.l Berndt, Fuss, and Waverman (1980) and Morrison and Berndt (1981) developed dynamic models in which capital is
quasi-fixed and subject to quadratic adjustment costs, but their approach utilizes an explicit solution to the optimal investment problem. In so doing it imposes the assumption that producers have static expectations regarding the evolution of
factor and output prices, and requires that the underlying cost function be
quadratic. Kennan (1979) and Meese (1980) estimated dynamic factor demand models in which producers have rational expectations, but Kennan imposed the restriction of a linear production structure, and Meese imposed a quadratic production
2 structure.
In an earlier paper (1982) concerned primarily with energy demand, we demonstrated an alternative approach that allows for a general production structure. It works as follows. In a stochastic environment, firms that have
rational expectations and maximize the expected sum of discounted profits also minimize the expected sum of discounted costs. Given any restricted cost function, one can derive the stochastic Euler equations (one for each quasi-fixed factor) that hold for this latter dynamic optimization problem. These Euler equations
-2-are just first-order conditions, and although they do not provide a complete
solution to the optimization problem, they can be estimated directly for any para-metric specification of the technology.
Our approach is to represent the technology by a translog restricted cost function, and then estimate the Euler equations, together with the cost function itself and the static demand equations for any flexible factors, using three-stage least squares. This permits us to test structural restrictions such as constant returns, and to test the over-identifying restrictions implied by
rational expectations. The estimated equations then provide a complete empirical description of the production technology, including both short-run (only
flexible factors adjust) and long-run (all factors fully adjust) elasticities of demand. The parameter estimates are fully consistent with rational expect-ations, and in particular with firm behavior that utilizes the solution to the underlying stochastic control problem.3
In our earlier paper we used U.S. manufacturing data for the period 1948 - 1971 to estimate a model in which capital and labor were treated as quasi-fixed factors, and energy and materials as flexible factors. We found adjustment costs on capital to be very important, but adjustment costs on labor appeared negligible. However, as we will soon see, this latter result was probably due to our aggregation of white-collar (skilled) and blue-collar
(unskilled) labor.
In this paper we utilize the same methodological approach described above, but both the model and data are different, and we focus on a number of different issues. In this paper we ignore the role of energy and materials, but we
disaggregate labor (into white-collar and blue-collar) and capital (into equip-ment and structures). This disaggregation turns out to be quite important; we find that white-collar labor has very significant adjustment costs, that adjustment costs on structures and equipment are quite different, and that structures and equipment enter the production structure differently. Also,
this paper studies the role of financing in more detail. In particular, we allow for both debt and equity financing, and estimate the extent to which
firms borrow to finance the marginal dollar of investment. Finally, we estimate the model with data for the period 1949 - 1976, thereby including those more recent years during which factor prices and rates of return fluctuated widely.
Our basic methodological approach is summarized in the next section. There we specify the translog restricted cost function (since blue-collar labor is
the only flexible factor this boils down to an input requirement function, and there are no static demand equations), and derive the stochastic Euler equations for the quasi-fixed factors. Section 3 briefly summarizes the estimation method, and discusses the treatment of various taxes and other issues related to the data. Estimation results are presented and discussed in Section 4.
2. The Models
Before presenting the details of our particular model specifications, it is useful to briefly review our general approach to estimating the production structure of a firm facing adjustment costs and making input choices in an uncertain environment. Let us assume that a time T the firm chooses levels of n variable inputs whose quantities and nominal prices are given by the vectors VT = (Vi ) and vT = (viT) respectively, and M quasi-fixed inputs whose
-- -T it iT
quantities are given by the vector X = (XiT). These inputs yield the single output Q. The technology can therefore be represented by a restricted cost function CT which specifies the minimum expenditure on variable factors needed to produce QT given the amounts of quasi-fixed factors X:
CT = C (VT, T' QT', ), (1)
with C increasing and concave in v but decreasing and convex in X, and the dependence on T capturing technical progress.
By definition, the firm incurs costs of adjusting the quasi-fixed factors. We assume these adjustment costs are convex and external to the firm, and we
4
-represent them in nominal terms by P h(AXT), where P is the price of output and AX = XT - X . The firm also makes direct outlays for its use of
quasi-T T T-l
fixed factors, and because of tax and financing considerations these expenditures may be spread through time. We therefore assume that outlays (net of adjustment
costs) for quasi-fixed factors at T, H, are a function of the current and past quantities of those factors:
HT = H (XT' "" X )-T) (2)
We assume the firm maximizes its expected present discounted value of profits. As shown in our earlier paper (1982), this implies that the firm minimizes the expected present discounted value of costs. Thus at time t the firm chooses a contingency plan for the vector of quasi-fixed factors to minimize:
00
min et R [C + Ph (AX ) + H] (3)
{X} - - t,T T T T
where t denotes the expectation conditional on information available at t, and
R 5
t, is the discount factor applied at t for costs incurred at .5
By taking the derivative of (3) with respect to Xil and setting it equal to zero, it is easily seen that the minimization yields the following Euler equations, or first-order conditions:
ac ah (AXt) ah(AXt+) + R
t it t axit t,t+l t+l axit j=0 tt+j 3xit
i = 1, ..., m (4)
These Euler equations just state that the net change in expected disounted costs from hiring one more unit of Xi at t is zero. That change is the sum of the increase in variable costs (which is negative), the extra costs of adjustment at t, the expected discounted value of the extra expenditures which the firm must incur by holding (at t only) one extra unit of Xi, minus the expected discounted
value of the savings in future costs of adjustment.
By Shepherd's Lemma we have the following additional first-order conditions, which take the form of static demand equations for the flexible factors:
(5) Vit = aCt / avit, i = 1, ..., n
As we discuss in more detail later, our approach is to estimate equations (1), (4), and (5) simultaneously.
In this paper we focus on four production inputs: equipment, structures, and white-collar and blue-collar labor. Our model pertains to the U.S. manu-facturing sector, which we treat as a single firm6 that takes input prices as given. Data for other inputs (e.g. energy and raw materials) were not available for the time period we consider, and we assume that such inputs are not sub-stitutable for labor and capital. In particular, we assume that expenditures on blue-collar workers (the only variable input) depend only on the wage of those workers, the levels of the other three inputs, and output.7
Since there is only one variable input, the restricted cost function (1) takes the form of an input requirement function, which we specify in translog form:
ln~t = a 0 a lnt a2lnEt+ a31nS + a4lnQt + Y 1 1(lnLt)
+ + 1 3 nLlnSt + 1lnLtlnEt 1 4lnLtlnQ + Y2 2(lnEt)2
+ y2 3lnEtlnSt + y2 41nEtlnQt + Y3 3(lnSt) + Y34 nStlnQt
+ ½y4 4(lnQt)2 + t (6)
where Bt, Lt, Et, St, and Qt are the levels of blue-collar labor, white-collar labor, equipment, structures and output at t. The term t allows for neutral technical progress. (The restrictions on the a's and y's required for this translog function to be decreasing and convex in Lt, Et and St are not imposed in the estimation.)
Costs of adjustment are assumed to be quadratic in ALt, AEt , and ASt.
In particular,
h = h (AL) + h (AE + ) ) + h(ASh(E t) = L(ALt) E(AEt) 2 + (AS
Note that cross effects -- i.e. changes in one factor affecting costs of
adjusting other factors -- are neglected. Letting wt and bt be the hourly wage
-
6-to white-collar and blue-collar workers respectively, the Euler equation for white-collar labor is then given by:
[bt(l-et)Bt/Lt ] (a1 + lllnLt + Y12 nEt + Y13nSt + Y14 nQt)
+ wt(1-et) + PtL (Lt-Lt 1) - Lt[Rt,t+lPt+l(Lt+-Lt)] = 0 (7) where 0 is the corporate income tax rate (wages are tax deductible in the U.S.).
The importance of tax and financing considerations becomes clear when we consider equipment. Let et be the purchase price of a unit of equipment. Suppose the firm buys a unit of equipment at t with the intention of keeping it until it is fully depreciated. If the firm borrows the present discounted value of the depreciation allowances its after-tax payment would be
et(l - Et - zEt ), where cEt is the investment tax credit on equipment and ZEt is the present discounted value of the depreciation allowance. If the
firm wants an extra unit of equipment at t without affecting the level of capital in subsequent periods, it will purchase (1-6E) fewer units of equipment at t+l, where 6E is the physical depreciation rate for equipment, thereby saving
(1-6E)et+l(1-CE,t+lZE,t+l). So far the only debt the firm incurs is offset
by the depreciation allowances. However, we would like to allow for the possibility that a fraction d of et(1-cEt-zEt) is borrowed for the purchase of the marginal
dollar's worth of equipment, and for simplicity we assume (perhaps unrealistically)
that this marginal debt is repaid after one year. (However, the firm is also
allowed to borrow unspecified amounts on inframarginal units of equipment.) Given this treatment of taxes and financing, and assuming that factors are productive in
the period in which they are purchased, the Euler equation for equipment is:
[bt(l-t)Bt/Et (12 2 + 12lnLt + 2 21nEt + Y2jlnS + Y24 tnQt)
+ (-d)et(-CEt-cEt ) + PtE 1 (Et-Et-1)
+
etRt,t+l
{d[l+it (1- t+l) ] e t (1 -C E t- z E t) - (-E)et+l ( 1- c Et + l-zE t+l)),
-
Pt+i
t+i-EE(E t) = (8)A similar analysis for structures yields the following Euler equation for that factor:
[bt(1-t)Bt/St (ca3+Y13lnLt+Y2 3lnEt+Y3 3lnSt+y341nQt) + (1-d)st(1-cSt-zSt)
+ P t (St-St_) 1 +etRt,t+l{d[l+it(l1- et+l) ]st (l- c St-zSt)
- (1-Cs)st+l(l s,t+lZS,t+1) - Pt+l(St+lSt)} = (9) where 6S, st, cst, and ZSt are, respectively, the physical depreciation rate,
the purchase price, the investment tax credit, and the present value of depre-ciation allowances for structures.
The model given by equations (6) - (9) is our "preferred" specification. One might argue, however, that the distinction between equipment and structures is somewhat artificial, since any capital that is bolted down is classified as a structure. We therefore estimate an alternative model for which equipment and structures are aggregated into a single measure of capital. The blue-collar
labor input requirement function is again specified to be translog:
lnB = + nL + K + 1nK + t 31nQt + 11(lnLt) + 12nLt nK
t 0 1 t 2 t 3 t 11 t 12 t t
+ 13nLtl + 22(lnK + 2 3 nKtlnQt + 13 3(lnQt) + Xt (10)
where Kt is the quantity of aggregate capital. Letting 6K kt, cKt, and ZKt denote, respectively, the physical depreciation rate, purchase price, investment tax credit, and present discounted value of depreciation allowances for this capital, the Euler equations are now given by:
[bt (1-8t)Bt/Lt] ( + 1 1lnLt + 1 2 nKt + ip1nQt3 ) + w t(1- ) + PtL(Lt-Ltl) - Lt[Rt,t+lPt+l(Lt+l-Lt)] = 0(11) [b t(It)Bt/K t] t 2 + 2 12 Lt + *22 nKt + *231nQt) + (l-d)kt(-CKt-Z Kt)1 + PtBK(Kt-Kt-i +
tRt,t+l
{d[l+it (l-t+)] kt(1-c Kt-zKt) - (1 Kt+l) -K)kt+l(1-C - Pt+lBK(Kt+i-Kt = (12) -f--PUsl-F I- 8-3. Estimation Method and Data
We obtain parameter values for "Model 1" by simultaneously estimating the input requirement function (6) and the Euler equations (7), (8), and (9), and for "Model 2" by simultaneously estimating the input requirement function (10) and the Euler equations (11) and (12). Note that because there is only one variable factor in each model, no static demand equations are estimated.
The estimation is done using three-stage least squares, which, if the error terms are conditionally homoscedastic, is equivalent to the generalized instru-mental variables procedure of Hansen (1982) and Hansen and Singleton (1982).11 That procedure minimizes the correlation between variables known at t (the
instruments) and the residuals of the estimating equations. The minimized value of the objective function of this procedure provides a statistic J, which is distributed as chi-squared, and which can be used to test the over-identifying restrictions of the model, as well as structural restrictions such as constant returns.
This instrumental variables procedure is a natural one to apply to the Euler equations, since the residuals of those equations can be viewed as expectational errors which, conditional on information available at t, have mean zero. The residuals of the input requirement function, however, cannot be interpreted
in the same way, since in theory that function should provide an exact description of the technology. In practice that function will not fit the data exactly,
and the errors are likely to be correlated with variables known at t. We will assume that this function holds in expectation with respect to some smaller conditioning set (i.e. set of instruments). We take that set to exclude current variables appearing in the input requirement function or Euler equations.
We estimate the models using annual data for the U.S. manufacturing sector for the years 1949-1976. Quantities and wage rates for blue- and white-collar labor are those compiled in Berndt and Morrison (1979). The purchase prices and quantities of equipment and structures were constructed by Ernst Berndt
using a perpetual inventory method, and assuming a physical depreciation rate for equipment of 0.135 and for structures of 0.071. Since no data for gross output in manufacturing is available for the period of our study, we used the
level of the gross domestic product of manufacturing from the National Income and Product Accounts. The price index Pt came from the same source.
We assume that the 1-period discount rate Rtt+l is equal to 1/(l+re), where r is the after-tax return on equity. 3 This return is constructed from
the identity:
re = rd(l-ep) + rc(1-) (13)
where rd and rc are respectively the dividend yield and capital gains rate (we use data on the Standard and Poor's 500 Index for both), p is the marginal personal tax rate (we use data reported by Seater (1980)), and is the effective tax rate on capital gains (based on the estimates of Feldstein and Summers
(1979), we set = 0.047). As for the interest rate i , we use the rate on commercial paper. The investment tax credit and present discounted value of
depreciation allowances are computed by Jorgenson and Sullivan (1981), who use
data on the term structure of interest rates to obtain z.
The following instruments are used in the estimation of both models: a constant, and the lagged detrended values of the rate of return on equity, the hourly compensation of both types of workers, the purchase prices of equipment and structures, the present discounted value of their depreciation allowances, as well as the logarithms of the quantities of blue-collar labor, white-collar
labor, structures, equipment, and output.
4. Estimation Results
For our "preferred" specification, both capital and labor are disaggregated, so that there are three quasi-fixed factors. Recall that the model for this specification ("Model 1") is given by the input requirement function (6) and the three Euler equations (7), (8), and (9). We also estimate the alternative
- 10
-fication in which equipment and structures are aggregated; this model ("Model 2") is given by the input requirement function (10) and the Euler equations (11) and (12). Each of these models was estimated first in its unrestricted form, and then with the restrictions of constant returns (CRTS) imposed.1 4
Parameter estimates are shown in Table 1. Both models satisfy the conditions of monotonicity and convexity for all but the first four years of data. Observe that for the unrestricted version of Model 1 the value of J is 54.45. Under the null hypothesis that the model is valid, J is distributed as chi-square with number of degrees of freedom equal to the number of instruments (13) times the number of equations (4) minus the number of parameters (20). The critical 5% level of the chi-square distribution with 32 degrees of freedom is 46.2, so that the over-identifying restrictions are rejected, throwing some doubt on the validity of the
15
estimated standard errors. For Model 2 the value of J is 34.81. For this model there are 39 - 14 = 25 degrees of freedom, the critical 5% level of the chi-square distribution is 37.7, and the over-identifying restrictions can be accepted.
To test for constant returns, we use the difference in the values of J with and without the parameter restrictions imposed. That difference is 418.22 - 54.45 = 363.77 for Model 1, and 447.83 - 34.81 = 413.02 for Model 2. There are five parameter restrictions in the CRTS version of Model 1, and four in Model 2, so that the critical 5% values of the chi-square distribution are 11.1 and 9.5 respectively. Constant returns is therefore overwhelmingly rejected.l6
The parameter estimates have interesting implications for the role of adjustment costs. Observe that all of the adjustment cost parameters have the
correct sign, and all are statistically significant except the one for structures in Model 1 (although this parameter is numerically large). The importance of adjustment costs is best understood by comparing their value as a total fraction of expenditures on a particular quasi-fixed factor with their value on the margin.
This is done in Table 2. In the first column of that table we take the average annual change in the stock of each quasi-fixed factor over the sample period, compute the adjustment cost for that average annual change using the parameter
(Asymptotic Standard Errors in Parentheses)
MODEL 1 (B; L,S,E) MODEL 2 (B; L,K)
Unrestricted -12.1166 (2.6060) 4.4941 (.3352) .00075 (.00090) -.000677 (.00082) .5907 (.9379) -.7502 (.0614) -5 3.010x10 (.00016) -.000103 (.000138) .3461 (.04815) .000191 (.000129) -.0002535 (9.624x10lO) -.000167 (.000194) .000563 (.000277) 6.181x10- 5 (.000173) -.5045 (.1782) -.01295 (.00302) 1.2733 (.09859) .000318_ (5.243x10-5) .00476 (.00238) .009238 (.009418) 54.45 .02186 1.181 .17229 1.727 .01858 1.776 EQ 9 SSR .01272 D.W. 2.771 CRTS 7.8492 (1.2135) -.7089 (.6326) -.001997 (.000541) -.001844 (.000652) .03281 (.1642) .000513 (.000123) .000352 (.000142) .000289 (.000113) -.000164 (7.878x10l5) -.000261 (.000104) -.02890 (.001186) 1.1938 (.09526) .000191 (.000189) .003159 (.002264) .02021 (.00868) 418.22 .06721 .737 3.2188 .218 .03566 .912 .02079 1.715 *0 *1 '2 2 *23 Unrestricted -10.9216 (2.3371) 4.5063 (.3342) -.003493 (.003411) .1395 (.8566) -.7527 (.06122) .000709 (.000611) .3473 (.04810) .000885 (.000656) -.001416 (.000659) -.4247 (.1620) d -.01213 (.002614) 1.5977 (.1752) .000315_ (5.246x10O ) .01768 (.003304) J EQ 10 SSR D.W. EQ 11 SSR D.W. EQ 12 SSR D.W. 34.81 .02154 1.160 .1717 1.708 .08443 1.647 CRTS 7.7565 (1.3809) -.7034 (.7172) -.01004 (.001362) .03160 (.1863) .002714 (.000347) .001593 (.00046) a0 a1 a2 a4 Yll Y14 Y22 Y23 Y33 ¥3 4 Y4 4 d OE
Bs
J EQ 6 SSR D.W. EQ 7 SSR D.W. EQ 8 SSR D.W. -.02427 (.000721) 1.4381 (.1635) .000185 (.000223) .01047 (.002753) 447.83 .09893 .506 3.2225 .217 .2643 .568 -- -- ---Y12- 12
-TABLE 2 - ADJUSTMENT COSTS
Percentage Marginal Percentage Total
Adjustment Cost (Avg.)* Adjustment Cost (Avg.)* Model 1 L .03 .001 E .23 .007 S .34 .005 Model 2 L .03 .001 K 2.14 .056
*Computations are explained in the text.
estimates in Table 1, and then divide by the average rental rate for the factor. This provides a measure of percentage marginal adjustment costs. In the second column we compute total adjustment costs for each factor in each year as a
fraction of the total expenditure on that factor, and then average these figures over the sample period. This provides a measure of percentage total adjustment costs. Observe that while adjustment costs are small as a total percentage of expenditures, they are quite large on the margin, particularly for capital. This means that the firm's cost minimization problem is very much a dynamic one.
Our estimates also have implications for the role of financing. In particular the parameter d specifies the fraction of the after-tax cost of a dollar of capital that is debt financed at the margin. However, our estimates of d are implausibly high since they are significantly greater than one for both models. While the parameter d applies only at the margin and thus the inframarginal investments may well lead to lower borrowing, these values seem high nonetheless.
Further insight into the production structure can be obtained by examining the elasticities of factor demands.17 These elasticities were calculated for the 1976 sample point, and are presented in Table 3.18
A number of things should be mentioned about these elasticities. First, note that there is consistency across the two models. Elasticities of blue- and white-collar labor demand with respect to the two wage rates and output are almost
identical across the two models, and in Model 2, the elasticities of demand for capital with respect to its own price and output are about midway between the cor-responding elasticities for equipment and structures in Model 1.
TABLE 3 - ELASTICITIES
Model 1 (B; L, S, E)
Long-run With respect to:
Elasticity of Demand for: P L E S PB PL PE PS B -1.4505 1.4505 -1.493x10- 5 1.734x10-5 .3575 L 2.3105 -2.3106 .000126 1.0007x10-5 .9938 E -.1554 .8166 -.5221 -. 1390 1.0582 S .2708 .09809 -.2107 -.1582 .5282
Short-run With respect to:
Elasticity of Demand for:
L E S Q
B -.6278 -9.7065xlo- 5 -6.4045xl0 -5 .98158
Model 2 (B; L, K)
Long-run With respect to:
Elasticity of Demand for: PB PL K B -1.4674 1.4676 -.000229 .3513 L 2.3381 -2.3385 .000394 1.0187 K -1.6851 1.8186 -.1335 .7275
Short-run With respect to:
Elasticity of Demand for:
L K Q
14
-Second, note that the elasticities are generally reasonable, both in magnitude and in sign. In Model 1, blue-collar labor and equipment are complementary inputs, as are structures and equipment, and in Model 2 blue-collar labor and the capital aggregate are complements. As one would expect, structures is relatively price inelastic, and in particular less elastic than equipment. The elasticities of demand for white-collar labor are large, but still reasonable.
Although we do not do so here, one could use the models presented in this paper to simulate the effects of changes in factor prices or output -- but only in a
deterministic context. As explained in Pindyck and Rotemberg (1982), such simu-lations are carried out by numerically solving the deterministic control problem that corresponds to the minimization in equation (3). A solution to the control problem can be obtained by finding initial conditions for the quasi-fixed factors which yield steady-state values for those factors (i.e. values which satisfy the associated transversality conditions) when the Euler equations and input require-ment function are together solved recursively through time. Note that for Model 1,
in which there are three quasi-fixed factors, this involves searching over a three-dimensional grid of initial conditions.
5. Conclusions.
The model of factor demands presented in this paper is consistent with rational expectations and dynamic optimization in the presence of adjustment costs. With the possible exception of the parameter which describes the manufacturing sector's
financing decisions, our estimates are quite plausible. We obtain reasonable elasti-city estimates, and find that equipment is a complementary factor to both blue-collar labor and structures, while other factor pairs are substitutes. As in our previous paper, we strongly reject constant returns to scale. Finally, we find that adjust-ment costs are very important at the margin, especially for equipadjust-ment and structures.
It is important to keep in mind that our model has a number of limitations, some of which are suggestive of further work. First, we aggregate across goods,
factors, and time. The numerous outputs produced under the heading of manufacturing are treated as a single good. Factor diversity is in fact richer that the two types of labor and capital we consider. Also, our use of annual data requires the implicit assumption that seasonal fluctuations are such that average output is only a function of average factor use over the year.
Second, our model imposes constraints on firms' financial policies. Although we allow for the different tax consequences of debt and equity financing, we ignore the term structures of the debt that firms incur when they purchase new capital. Finally, we only allow for limited forms of technical progress. Expanding the model to include other forms of technical progress might significantly affect the parameter estimates.
16 -FOOTNOTES
1. As Lucas (1967) and Treadway (1971) have shown, under certain conditions the flexible accelerator is consistent with dynamic optimization in the presence of adjustment costs. For a survey of this and related models, together with an assessment of their empirical performance, see Clark (1979).
2. For a survey of some of the recent work in dynamic factor demand modelling, see Berndt, Morrison, and Watkins (1981).
3. Since we do not actually solve the stochastic control problem (beyond writing the first-order conditions), we cannot calculate optimal factor demand trajec-tories corresponding to particular stochastic processes for prices. Stochastic control problems of this sort are generally difficult, if not impossible to solve, and this raises the question of whether rational expectations provides a realistic behavioral foundation for studying investment behavior and factor
demands in general.
4. As in Gould (1968). For a survey of the treatment of adjustment costs, see Soderstrom (1976). We implicitly assume that costs of adjustment are incurred only when the net quantity of X changes. This assumption was relaxed in our earlier paper (1982), where our point estimates suggest that, indeed, only changes in Xit lead to adjustment costs. We also estimated the current models relaxing this requirement, without affecting our results significantly.
5. Note that while the expectation in (3) treats not just future input prices but also future output as random variables, output is not predetermined. The ran-dom variable Q is given by the contingency plan that maximizes expected profits. 6. Or, equivalently, as consisting of many competitive firms whose aggregate
technology is given by our model.
7. This assumption is justified if the production function is of the Leontief form in two composite inputs, the first of which is a function of the levels of labor and capital while the second is a function of the other inputs.
8. We also estimated versions of the model assuming adjustment costs are quadratic in ALt/L t_ AEt/Et-1' and ASt/Sti, without significantly affecting the results. 9. This assumes the firm purchases some equipment at t+l. If the firm bought an
extra unit of equipment at t with the intention of selling (1-6E) units at t+l, it would be unlikely to borrow the expected present value of the depreciation allowances. Instead it would pay et(l-t-zEt) at t, where ZEt is the
depreci-t Et Et Et
ation allowances in the first period, and it would receive (1-6E)et+l(l-CE ,t+l) at t+l.
10. If revenues at t+l get discounted at t at the rate it(1-et+l), Rtt+l
1/(l+it(1-et+l)), the fraction d that is debt financed is irrelevant to the firm.
11. This equivalence is explained in our earlier paper (1982).
12. These data incorporate some embodied technical progress since employment levels are corrected for the educational achievement of the work force.
13. We also estimated the models using Rt t+l = l/(l+rcp), where rp is the after-tax return on commercial paper, but we obtained significantly poorer results. 14. But note that the constant returns version is estimated using the covariance
matrix obtained from the estimation of the unrestricted model, thereby per-mitting us to test the CRTS restrictions. CRTS parameter restrictions are derived as in Pindyck and Rotemberg (1982).
15. Failure of the over-identifying restrictions in this model is inconsistent with the hypothesis that firms are optimizing with rational expectations. 15. Note that this puts into question empirical q-theory models of investment
that equate marginal and average q.
17. Intermediate- and long-run elasticities, i.e. those that apply when quasi-fixed factors have partially or fully adjusted, must be interpreted with caution. The reason is that if prices evolve stochastically, the adjustment path for any particular discrete change in a price, as well as the long-run expected
(continued) ____ ____ _______111__111 111
__(·^__·_··· 18 __(·^__·_···
-equilibrium, are solutions to a stochastic control problem (and in some cases a long-run expected equilibrium may not exist). Since such solutions are typically infeasible, we must compute elasticities by implicitly assuming that firms ignore the variance of future prices in responding to price changes. These elasticities can therefore be best viewed as a description of the technology.
18. For an explanation of how these elasticities are calculated, see the Appendix of Pindyck and Rotemberg (1982).
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